Question

Show that there are infinitely many positive primes.
pls ans this question ​

Answer 1

Given : Prime number

To Find  : Show that there are infinitely many positive primes.

Solution:

Let say there are finite prime numbers

P₁ , P₂ , . .   .  , Pₙ

x = 1 +  P₁ P₂. . .   .  .Pₙ

P₁ P₂. . .   .  .Pₙ  is divisible by all primes P₁ , P₂ , . .   .  , Pₙ

hence x = 1 +  P₁ P₂. . .   .  .Pₙ  is not divisible by any of P₁ , P₂ , . .   .  , Pₙ

=>  x is a prime number

or x has prime divisor other than P₁ , P₂ , . .   .  , Pₙ

This contradicts our assumptions

Hence we can sat prime numbers are not finite

and there  are infinitely many positive primes

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